Parameter degeneracies and (un)predictability of gravitational microlensing events

(abridged) Some difficulties in determining the physical properties that lead to observed anomalies in microlensing light curves, such as the mass and separation of extra-solar planets orbiting the lens star, or the relative source-lens parallax, are already anchored in factors that limit the amount of information available from ordinary events and in the adopted parametrization. Moreover, a real-time detection of deviations from an ordinary light curve while these are still in progress can only be done against a known model of the latter, and such is also required for properly prioritizing ongoing events for monitoring in order to maximize scientific returns. Despite the fact that ordinary microlensing light curves are described by an analytic function that only involves a handful of parameters, modelling these is far less trivial than one might be tempted to think. A well-known degeneracy for small impacts, and another one for the initial rise of an event, makes an interprediction of different phases impossible, while determining a complete set of model parameters requires the assessment of the fundamental characteristics of all these phases. While the wing of the light curve provides valuable information about the time-scale that absorbs the physical properties, the peak flux of the event can be meaningfully predicted only after about a third of the total magnification has been reached. Parametrizations based on observable features not only ease modelling by bringing the covariance matrix close to diagonal form, but also allow good predictions of the measured flux without the need to determine all parameters accurately. Campaigns intending to infer planet populations from observed microlensing events need to invest some time into acquiring data that allows to properly determine the magnification function.Comment: 6 pages with 4 EPS figures embedded; MNRAS accepte

The binary gravitational lens and its extreme cases

The transition of the binary gravitational lens from the equal mass case to small (planetary) mass ratios q is studied. It is shown how the limit of a (pure shear) Chang-Refsdal lens is approached, under what conditions the Chang-Refsdal approximation is valid, and how the 3 different topologies of the critical curves and caustics for a binary lens are mapped onto the 2 different topologies for a Chang-Refsdal lens with pure shear. It is shown that for wide binaries, the lensing in the vicinity of both lens objects can be described by a Taylor-expansion of the deflection term due to the other object, where the Chang-Refsdal approximation corresponds to a truncation of this series. For close binaries, only the vicinity of the secondary, less massive, object can be described in this way. However, for image distances much larger than the separation of the lens objects, any binary lens can be approximated by means of multipole expansion, where the first non-trivial term is the quadrupole term. It is shown that an ambiguity exists between wide and close binary lenses, where the shear at one of the objects due to the other object for the wide binary is equal to the absolute value of the eigenvalues of the quadrupole moment for the close binary. This analysis provides the basis for a classification of binary lens microlensing events, especially of planetary events, and an understanding of present ambiguities.Comment: 20 pages in LaTeX2e format with 9 embedded PostScript figures; figures modified and embedded; accepted for publication in A&

Stochastical distributions of lens and source properties for observed galactic microlensing events

A comprehensive new approach is presented for deriving probability densities of physical properties characterizing lens or source that constitute an observed galactic microlensing event. While previously encountered problems are overcome, constraints from event anomalies and model parameter uncertainties can be incorporated into the estimate. Probability densities for given events need to be carefully distinguished from the statistical distribution of the same parameters among the underlying population from which the actual lenses and sources are drawn. Using given model distributions of the mass spectrum, the mass density, and the velocity distribution of Galactic disk and bulge constituents, probability densities of lens mass, distance, and the effective lens-source velocities are derived, where the effect on the distribution that arises from additional observations of annual parallax or finite-source effects, or the absence of significant effects, is shown. The presented formalism can also be used to calculate probabilities for the lens to belong to one or another population and to estimate parameters that characterize anomalies. Finally, it is shown how detection efficiency maps for binary-lens companions in the physical parameters companion mass and orbital semi-major axis arise from values determined for the mass ratio and dimensionless projected separation parameter, including the deprojection of the orbital motion for elliptical orbits. Compared to the naive estimate based on 'typical values', the detection efficiency for low-mass companions is increased by mixing in higher detection efficiencies for smaller mass ratios (i.e. smaller masses of the primary).Comment: 25 pages with 22 embedded EPS-figures, uses mn2e.cls. Adopted mass function revised and one example event replaced, content rearranged, some minor changes. Submitted to MNRA

Astrometric Resolution of Severely Degenerate Binary Microlensing Events

We investigate whether the "close/wide" class of degeneracies in caustic-crossing binary microlensing events can be broken astrometrically. Dominik showed that these degeneracies are particularly severe because they arise from a degeneracy in the lens equation itself rather than a mere "accidental" mimicking of one light curve by another. A massive observing campaign of five microlensing collaborations was unable to break this degeneracy photometrically in the case of the binary lensing event MACHO 98-SMC-1. We show that this degeneracy indeed causes the image centroids of the wide and close solutions to follow an extremely similar pattern of motion during the time when the source is in or near the caustic. Nevertheless, the two image centroids are displaced from one another and this displacement is detectable by observing the event at late times. Photometric degeneracies therefore can be resolved astrometrically, even for these most severe cases.Comment: 11 pages, including 4 figures. Submitted to Ap